Partitions of the Set of Prime Numbers and Maximal Sequences of Consecutive Primes

For each digit i ∈ {0, …, 9}, we consider the partition of the set of primes P into two complementary classes: the class of primes whose decimal representation does not contain the digit i, denoted {Pi⁻}, and the class of those that contain it at least once, denoted {Pi⁺}. We will first establish that every set {Pi⁻} has zero density in P, which ensures that its complement {Pi⁺} is infinite in P. This property will then be generalized to any set of s digits, where 1 ≤ s ≤ 9. Building on this result and on J. Maynard’s 2019 theorem ([2]), which guarantees the infinity of {Pi⁻}, we will prove that the sets {Pi⁺} contain arbitrarily long sequences of consecutive primes. We then present the results of a systematic numerical exploration up to 10¹¹, which lead us to a conjecture regarding the boundedness of sequences of consecutive primes in {Pi⁻}. We observe a clear separation of the maximum lengths of these sequences into two families, depending on whether the digit i can (family A) or cannot (family B) appear in the last position of a prime number. The record for length is held by a sequence of 117 consecutive primes excluding the digit 4 (family B), while the shortest is a sequence of 37 consecutive primes excluding the digit 3 (family A). For any i, at least in the interval [2, 10¹¹], we note the completely different behavior between the maximal sequences of {Pi⁺} and the maximal sequences of {Pi⁻}, these two complementary sets being disjoint and infinite sets.